2012 AMC 10B Problem 19

Question

In rectangle

ABCD,

AB=6,

AD=30,

and

G

is the midpoint of

\overline{AD}.

Segment

AB

is extended

2

units beyond

B

to point

E,

and

F

is the intersection of

\overline{ED}

and

\overline{BC}.

What is the area of quadrilateral

BFDG?

\dfrac{133}{2}

67

\dfrac{135}{2}

68

\dfrac{137}{2}

Use similar triangles

EBF

and

EAD

Quadrilateral

BFDG

is a trapezoid with height

6

The polygon

BFDG

is a trapezoid with bases

DG

and

BF

and height

6.

Also, since

G

is the midpoint between

A

and

D,

we have

GD= 15.

We can see that

EBF \sim EAD ,

so

\dfrac{BF}{AD} = \dfrac{EB}{EA}

\dfrac{BF}{30} = \dfrac 28

BF = 7.5

This makes the area of

BFDG

equal to

\dfrac{6(15+7.5)}2 = \dfrac{135}2.

Thus, the correct answer is C .

Problem 19 in other years